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arXiv · 1607.01655

L1 penalization of volumetric dose objectives in optimal control of PDEs

Richard C. Barnard·Christian Clason
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Abstract

This work is concerned with a class of optimal control problems governed by a partial differential equation that are motivated by an application in radiotherapy treatment planning, where the primary design objective is to minimize the volume where a functional of the state violates a prescribed level, but prescribing these levels in the form of pointwise state constraints can lead to infeasible problems. We therefore propose an alternative approach based on $L^1$ penalization of the violation. We establish well-posedness of the corresponding optimal control problem, derive first-order optimality conditions, and present a semismooth Newton method for the efficient numerical solution of these problems. The performance of this method for a model problem is illustrated and contrasted with the alternative approach based on (regularized) state constraints.

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Richard C. Barnard, Christian Clason. 2016-07-06. L1 penalization of volumetric dose objectives in optimal control of PDEs. https://doi.org/10.1007/s10589-017-9897-6

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